Construction of Combinatorial Objects 1. a General Point of View

Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classiied. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all nite abelian groups and with much more eeort all nite simple groups. More often one restricted the research activity rstly to more modest problems like the pure existence of any object with some prescribed properties, for example in the case of block-designs or even when solutions for some optimization problem are considered. A step further was taken in combinatorics where the number of objects was determined by ingenious methods without any relation to a direct construction idea. For example the famous PP olya-De Bruijn method of counting orbits of a group acting on sets of mappings allowed to determine exact or approximate numbers of diierent types of graphs, e.g. especially from a more practical point of view, the number of diierent chemical isomers of a certain type 57, 11, 33]. Now that scientists have more and more powerful computers available in some nontrivial cases the construction problem itself can be attacked successfully. Thus, some of the above mentioned nite simple groups have been constructed by the help of a computer, where the theoretical approach gave very restrictive necessary conditions for the existence of some sporadic simple group. With respect to the mathematical description of chemical structures to a certain extent the above mentioned counting approach could be replaced by a construction process. The present paper aims at an introduction into some general methods for construction algorithms. Concrete results for special types of objects will serve as an example and hopefully give an impact for further applications in diierent areas. We restrict our attention to isomorphism problems. More formally we assume that a group is acting on a set of objects such that the orbits are just …